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A locally constant sheaf on a locally connected space is a covering space; Proof?
As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here ...
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1answer
25 views
What subsets of a covering space cover their image?
Say I have a covering map $p \colon E \to B$. Then for which subsets $F$ of $E$, is $p|_F \colon F \to p(F)$ a covering map?
If it makes things easier, assume $E$ is simply connected, that is, the ...
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48 views
Covering infinite sheeted covering of torus
Suppose I have subgroup $H=\operatorname{span}\langle (a,b)\rangle\subset \pi_1(\mathbb{T}^2)=\mathbb{Z}^2$, where $a,b$ are integers where $(a,b)\neq(0,0)$. I know the covering space is $S^1\times ...
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preimage of a connected under a covering map has unique representation into slices
Let $p:E\to B$ be a covering map. Suppose that $U$ is an open set of $B$ that is evenly covered by p. Show that if $U$ is connected, then the partition of $p^{-1}(U)$ into slices is unique.
I have no ...
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89 views
The definition of normal covering in Hatcher book
In page 70 of Hatcher's book, in the section Deck Transformations and Group Actions, the author defines a normal covering in the following way:
A covering space $p:\tilde X\to X$ is called normal ...
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function lifting on $S^1 \times S^1$
Let $f:S^1 \times S^1 \to S^1 \times S^1$ a continuous function and $p:\mathbb{R}^2 \to S^1 \times S^1: (t,s) \mapsto (e^{2\pi i t},e^{2\pi i s})$ a covering map. if $F: \mathbb R ^2 \to \mathbb R ^2 ...
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Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes
Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$.
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909 views
The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent
This is exercise 1.3.8 in Hatcher:
Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then ...
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An entire function with finite covering group is a polynomial.
Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
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Constructing Riemann surfaces using the covering spaces
In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow:
A polynomail-like map ...
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Algorithms for covering a rectilinear polygon using rectangles of the same size
The following is the problem description:
All angles of the polygon are right. It may be convex or concave. Use rectangles of the same size to cover the polygon. The edge of the polygon and ...
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A good way to understand Galois covering?
A covering map $f:X\rightarrow Y$ is called Galois if for each $y\in Y$ and each pair of lifts $x, x^{'}$, there is a covering transformation taking $x$ to $x^{'}$. What is a good way to understand ...
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How to construct certain cover given in Mumford's Abelian Varities book
In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in ...
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Number of sheets of Covering space of $\mathbb{T}^2$ after transformation by $SL(2,\mathbb{Z})$
Suppose you have a subgroup of $H\subset\pi_1(\mathbb{T}^2)=\mathbb{Z}\times\mathbb{Z}$, $H=\text{span}\langle u,v\rangle$. If you have an element $G\in SL(2,\mathbb{Z})$, do the covers of ...
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How do we check if a covering of an orbifold is a manifold?
Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
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Example of a nontrivial finite covering map
A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
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Characterization of maps with $\mathbb{Z}/2\mathbb{Z}$-equivariant lifts.
I'm interested in characterizing maps $f:\mathbb{R}P^k\to\mathbb{R}P^\infty$ that lift to a $\mathbb{Z}/2\mathbb{Z}$-equivariant map $\tilde{f}:S^k\to S^\infty$.
For $k\geq 2$ I have been able to ...
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Are there generalizations of Prym varieties to higher dimensions
Prym varieties are abelian varieties that are associated to a double cover of algebraic curves.
Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way?
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42 views
Global sections of covering spaces
Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$.
Is there any reference where this is explicitly ...
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Identifying the numbers of degree $n$ covering spaces of $X$
Let $X$ be a path-connected, locally path-connected and semilocally simply-connected space. Can we find a correspondence between degree $n$ covering spaces of $X$ and group homomorphism ...
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Monodromy Theorem and Homotopy Lifting Theorem
I've just come across this proof of the following theorem that I can't convince myself is true. Any ideas whether it's correct?
Suppose $\gamma$ and $\lambda$ are homotopic paths starting at $x$ in a ...
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52 views
Liftings of curves $u\cdot v$ and $v\cdot u$ with respect to the sine covering map.
I'm trying to work through the exercises in Otto Forster's book on Riemann Surfaces. While most of them seemed not that hard, this one gives me a headache:
Let $X=\mathbb{C}\setminus\{\pm1\}$ and $Y ...
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45 views
What are the branch points of $X(n)\to X(1)$
Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps).
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Hyperbolic Universal Covering Space
I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic ...
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Why is the pullback of a connected cover not necessarily connected?
In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
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The covering space of a region contained in complex plane delete two points.
We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic ...
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Degree of morphism of quotient of upper half-plane
Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
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Question on homotopy lifting
I'm studying covering maps and homotopy lifting and I would like to clarify a few things which my lecture notes doesn't seem to make clear.
A lemma in my lecture notes says:
Let $p: \tilde Y \to ...
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Classification theorem of the coverings of a given space
I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as
Classification of covering spaces ...
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How to find every 4-sheet covering of the wedge sum?
Based on this question 4-sheet covering of the wedge sum of two circles I know how to find one 4-sheet covering of the wedge sum, but how to find every 4-sheet covering of the wedge sum?
I really ...
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52 views
How do I prove that a map is not a covering map?
I'm thinking how to prove that a map is not a covering map. For example let $p:\mathbb R_+\to S^1$ be a map defined by $p(\theta)=(\cos(2\pi\theta),\sin(2\pi\theta))$. I'm trying to find a point which ...
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Are there infinitely many rational functions of bounded degree and given ramification
It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus is finite (up to isomorphism).
Edit. The branch locus $B$ of $f:X\to ...
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Completelly cover area with minimum number of maxed circles NP-completeness (or harder) proof
everyone.
I'm looking for paper with proof of NP-completeness following, or similar problem.
Given:
Area $S \subset \mathbb{N}^2$, let it be convex or rectangular, I believe it doesn't matter
...
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75 views
Universal cover as a principal $\pi_1$ bundle.
Let $M$ be a connected manifold with universal cover $\tilde M$ and fix $x_0 \in M$. Then it is well-known that $\tilde M \to M$ is a principal $\pi_1(M,x_0)$ bundle. I'm a bit confused about the ...
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What is a “mere cover”?
Sorry to ask such a basic question, but I'm having a lot of trouble finding a definition of this. I saw this term in Stefan Wewers' thesis and it seemed familiar, but googling "mere cover" doesn't ...
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Covering space of a graph is again a graph - why??
i want to prove the statement in the heading. Thus given graph $G$ and a covering space $p:\tilde{X}\rightarrow G$ to prove that $\tilde{X}$ is also a graph. My idea was to take $\tilde{E}=p^{-1}(E)$ ...
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Which of the following spaces nontrivially cover themselves?
I am having some difficulties with a qualifying exam question. I would appreciate if someone could give me a little help.
Which of the following spaces nontrivially cover themselves?
(a) $S^3$
(B) ...
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38 views
Compactness of covering space
If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is?
Torus or sphere, make me believe the answer is yes.
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Restriction of a covering map to a subspace
Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map.
Hint: Show first that if $V\subset Y$ is well-covered by $p$, then ...
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Pullback of a family of curves via a covering map.
Let $X$ be a smooth compact projective manifold and $\pi:Y\rightarrow X$ a Galois covering map. Is it always possible to pull back a family of curves on $X$ to $Y$?
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Covering space, Weyl group, flag manifold.
Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
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Covering space (Lie groups and their maximal tori)
Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
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Very Basic Covering Space Q. - Two different covering maps over a same space?
Here's the situation I'm in.
I have a map from the (closure of) Upper Half Plane ($\mathbb{U}$) into the punctured (closed) disk ($\mathbb{D}$) called $q$ that satisfies
$q(0) = 1$, and $q(z+1) = ...
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What is the Hurwitz number of an elliptic curve
One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points.
Suppose that $X$ is an elliptic ...
